Hi,I have read probably hundreds of reviews on several pages but I am still not quite sure what I should buy.

Purpose of this new camera is especially to take stills and movies of our half-year old son and also to take pictures from holidays (mainly landscape, architecture, etc). At this moment I have at home Sony Alpha 100 (old amateur DSLR) which struggles especially in low light (under bulb lighting in flat). To my surprise most pictures from Alpha look worse then images from my HTC One phone. So we want something which will be able to replace it. Full disclosure is also that this will be main camera in our house. Camera will be used both by me and my wife - she will be shooting auto only I suppose. So good IQ of camera is important. Also big priority is on portability because I do no longer enjoy carrying DSLR with bag. So this new camera should be ideally pocketable in coat or in light small bag. Although I am also considering mirrorless I do not expect to buy many new lenses except if there would be really great deals. So premium primes are out of our possibilities at this moment, so camera should shoot nicely with kit lens. Based on below list you will see that our budget is around equivalent of 700-800 USD, I am also adding prices from our country(Czech Republic member of EU) so you will see that prices are much higher than in USA however import is out of question because on shipping, taxes, custom fees difference would be minimal if any.

Here is list of cameras I have so far pre-chosen and some comment why.1) Advanced compacts:Olympus XZ-2 - it has won or was very good in several tests I have read. It has fast lens, should have nice handling and price is around 400 USDSony RX100 - from most tests this and mark II seems to be ultimate compacts. In Czech is available for approximately 600-700 USD, mark 2 is however too expensive here around 900 USD which is too expensive for me, or at least it seems to me so because it is only compact and I am not sure wheter I would use hotshoe at all.Canon S120 - as advantage I see that it is more versatile with bigger zoom and most pocketable. price is between 500-600 USD

2) then based on reviews I have started to think about 4/3 formatOlympus E-PL5 or 6 - I have read many good reviews. so far only criticism I have read that Olympus is generally inferrior in movie category (is it so bad?) It can be bought here for around 750 USDOlympus E-PM2 - smaller and cheaper then E-PL5. So far I understand that image quality is same, difference is only in size and handling. this is here cheaper for around 600 USD. If I will chooose Olympus I will probably decide between E-PL5 and E-PM2 based on handling in shopPanasonic DMC-GM1 - I like look of this camera and I am impressed by its size. However it is also most expensive at around 900 USD

3) I was almost decided to take either Sony RX-100 or some 4/3 format but then started to be concerned that most reviews say that they provide great quality however as it is expected there is usually little bit less detail and there is worser low-light performance than from APS-C format cameras and also none of them has phase detection which should be good for shooting of child once he starts to crawl and move more. Based on that I have added to list:Sony 5T - should have phase detection and overall great image quality and size is still acceptable to me. Price around 800 USD Samsung NX-300 - again phase detection and body seems to be nice in hand. Price around 700 USD

4) last category I have added is Nikon 1 system which is probably inferior in many areas, however it should offer quick autofocus including phase and older models are very cheap.Nikon 1v1 - available here for around 350 USD, bad thing is that flash is not included and sold for another 170 USDNikon 1J1 - available for as low as 260 USDother models like 1j3 are already in range of 600 USD which seems to be quite match for it

Last I will add that we print individual photos in 4x6 and my wife is also creating photo-books which are then printed in A4 format. So occasionally we can use individual photo of A3 size (when it is printed on both sides of book or on cover) but this is less likely.

I ask advice mainly because I am now confused whether I should really seek camera with phase detection for shooting of kid (when he will be older and faster) or if generally for this type of photos is 4/3 format still ok especially with in many reviews acclaimed quick autofocus of Olympus or Panasonic. And whether APS-C format advantage in low light is so high. Or if in case I am not looking to invest into expensive lenses I may as well be better served with advanced compact. Or whether Nikon 1 system is not so bad and at these bargain prices should be considered seriously?

I know that this is difficult question and I have provided many options but based on fact that I am no expert in photography I will really appreciate every advise

For shooting moving kids, you probably need C-AF. I would choose a previous generation DSLR (to keep costs down) or the Nikon J1, which I believe is a better mirrorless for sports (and moving kids). Those Nikons are very small, but I've tried one and didn't like the shooting experience at all.

As the sensor size gets smaller, critically-accurate focusing becomes less of an issue because of the deeper depth of field. You can still shoot moving kids with something like the E-PM2 because of its fast focusing, but it would take a bit of practice.

Many of those advanced compacts are a bit of luxury really, because you can get previous generation mirrorless models at lower prices with better IQ.

Or if in case I am not looking to invest into expensive lenses I may as well be better served with advanced compact.

I would agree with that sentiment, so I would recommend a Sony RX100. (Full disclosure: I own one.) Just keep in mind that, when shooting indoors, you'll probably have to leave the lens at its widest setting and "zoom with your feet", as the saying goes. (Also, note that the RX100 has the same size sensor as the Nikon 1 series, so there's no advantage to the Nikon unless you buy other lenses, which you've decided not to do.)

Then, in 3-5 years or so, upgrade to something else (maybe with hybrid contrast/phase detection AF which by then should be much more common*, and likely cheaper than it is now) since you'd probably be looking to upgrade around then anyway, even if you bought a 4/3 camera now.

And lastly, re: "And whether APS-C format advantage in low light is so high." -- as someone who owns both an Olympus E-PL5 and Canon 70D (with an APS-C sensor), I would say no, there's not that much difference between the two camera bodies. As usual, with interchangeable lens systems, it's more about which lenses you use them with. And with comparable lenses mounted, the E-PL5 is much more portable than the 70D. And since portability is an important criteria to you, I would think that would outweigh only slightly better low-light performance.

Mark

*note that your statement: "4/3 format but ... none of them has phase detection" is incorrect, as the Olympus E-M1 has phase detection. Not that you should consider buying an E-M1 now since it's over your budget, but that's why I recommended waiting and upgrading in a few years...